# The space-time fractional diffusion equation with Caputo derivatives

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## Abstract

We deal with the Cauchy problem for the space-time fractional diffusion equation, which is obtained from standard diffusion equation by replacing the second-order space derivative with a Caputo (or Riemann-Liouville) derivative of order β∈(0, 2] and the first-order time derivative with Caputo derivative of order α∈(0, 1]. The fundamental solution (Green function) for the Cauchy problem is investigated with respect to its scaling and similarity properties, starting from its Fourier-Laplace representation. We derive explicit expression of the Green function. The Green function also can be interpreted as a spatial probability density function evolving in time. We further explain the similarity property by discussing the scale-invariance of the space-time fractional diffusion equation.

## AMS Mathematics Subject Classification

26A33 49K20 44A10## Key words and phrases

Fractional diffusion equation time-space Caputo derivative Fourier transform Laplace transform Mittag-Leffler function Green function stable probability distributions## References

- 1.O. P. Agrawal,
*Solution for a fractional diffusion-wave equation defined in a bounded domain*, Nonlinear Dynamics**29**(2002), 145–155.MATHCrossRefMathSciNetGoogle Scholar - 2.V. V. Anh and N. N. Leonenko,
*Non-Gaussian scenarios, for the heat equation with singular initial conditions*, Stochastic Processes and their Applications**84**(1) (1999), 91–114.CrossRefMathSciNetGoogle Scholar - 3.V. V. Anh and N. N. Leonenko, Scaling laws for fractional diffusion-wave equations with singular data, Statistics and Probability Letters Volume
**48**(3), (2000), 239–252.MATHCrossRefMathSciNetGoogle Scholar - 4.V. V. Anh and N. N. Leonenko,
*Spectral analysis of fractional kinetic equations with random data*, J. Stat. Physics,**104**, N5/6 (2001), 1349–1387.MATHCrossRefMathSciNetGoogle Scholar - 5.V. V. Anh and N. N. Leonenko,
*Renormalization and homogenization of fractional diffusion equations with random data*, Probab. Theory Rel. Fields**124**(2002), 381–408.MATHCrossRefMathSciNetGoogle Scholar - 6.V. V. Anh and N. N. Leonenko,
*Harmmonic analysis of fractional diffusion-wave equations*, Applied Math. Comput.**48**(3) (2003), 239–252.MathSciNetGoogle Scholar - 7.M. Basu and D. P. Acharya,
*On quadratic fractional generalized solid bi-criterion*, J. Appl. Math. & Computing (old: KJCAM)**10**(2002), 131–144.MATHMathSciNetGoogle Scholar - 8.M. Caputo,
*Linear model of dissipation whose Q is almost frequency indepent-II*, Geophys. J. R. Astr. Soc.**13**(1967), 529–539.Google Scholar - 9.M. Caputo,
*The Green function of the diffusion of fluids in porous media with memory*, Rend, Fis. Acc. Lincei (Ser. 9)**7**(1996), 243–250.MATHCrossRefGoogle Scholar - 10.M. M. Djrbashian,
*Integral transforms and representations of functions in the complex plane*, Nauka, 1966 (in Russian).Google Scholar - 11.A. M. A. El-Sayed and M. A. E. Aly,
*Continuation theorem of fractionalorder evolutionary integral equations*, J. Appl. Math. & Computing (old: KJCAM)**9**(**2**) (2002), 525–534.MATHMathSciNetGoogle Scholar - 12.A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi,
*Higer transcendental functions*, New York, McGraw-Hill Vol. 3, 1953-1954.Google Scholar - 13.W. Feller,
*On a generalization of Marcel Riesz's potentials and the semigroups generated by them*, Meddelanden lunds Universitets Matematiska Seminarium (Comm. Sém. M..a.thém. Université de Lund), Tome suppl. dédié à M. Riesz, Lund, pp. 73–81, 1952.Google Scholar - 14.Y. Fujita,
*Integrodifferential equation which interpolates the heat equation and the wave equation*, Osaka, J. Math.**27**(1990), 309–321.MATHMathSciNetGoogle Scholar - 15.A. A. Kilbas, T. Pierantozzi and J. Trujillo,
*On the solution of fractional evolution equations*, J. Phys. A: Math. Gen.**37**(2004), 3271–3283.MATHCrossRefMathSciNetGoogle Scholar - 16.R. Gorenflo, Yu. Luchko and F. Mainardi,
*Wright function as scale-invariant solutions of the diffusion-wave equation*, J. Comp. Appl. Math.**118**(2001), 175–191.CrossRefMathSciNetGoogle Scholar - 17.R. Gorenflo and F. Mainardi,
*Fractional calculus: integral and differential equations of fractional order*, in A. Carpinteri and Mainardi (Editors), Fractals and Fractional Calculus in Continuum Mechanics, Wien and New York, Springer Verlag., pp. 223–276, 1997.Google Scholar - 18.R. Gorenflo and F. Mainardi,
*Random walk models for space-fractional diffusion processes*, Fractional Calculus and Applied Analysis**1**, (1998), 167–191.MATHMathSciNetGoogle Scholar - 19.R. Gorenflo and F. Mainardi,
*Approximation of Lévy-Feller diffusion by random walk*, Journal for Analysis and its Applications (ZAA)**18**(1999), 231–246.MATHMathSciNetGoogle Scholar - 20.R. Gorenflo and F. Mainardi, D. Moretti, G. Pagnini,
*Time-fractional diffusion: a discrete random walk approach*, Nonlinear Dynamics**29**(2002), 129–143.MATHCrossRefMathSciNetGoogle Scholar - 21.R. Gorenflo and F. Mainardi, D. Moretti and G. Pagnini,
*Disctete random walk models for space-time fractional diffusion*, Chemical Physics**284**(2002), 521–541.CrossRefGoogle Scholar - 22.R. Hilfer,
*Exact solutions for a class of fractal time random walk*, Fractals**3**(1995), 211–216.MATHCrossRefMathSciNetGoogle Scholar - 23.F. Huang and F. Liu,
*The time fractional diffusion equation and advection-dispersion equation*, the Australian and New Zealand Industrial and Applied Mathematic Journal (ANZIAM), (2004), in press.Google Scholar - 24.F. Liu, V. V. Anh, I. Turner and P. Zhuang,
*Time Fractional Asvection-dispersion Equation*, J. Appl. Math. & Computing**13**(2003), 233–245.MATHMathSciNetCrossRefGoogle Scholar - 25.F. Liu, V. V. Anh and I. Turner,
*Numerical, solution of the space fractional Fokker-Plank Equation.*, J. Comp. Appl. Math**166**(2004), 209–219.MATHCrossRefMathSciNetGoogle Scholar - 26.F. Mainardi,
*Fraction calculus: some basic problems in continuum and statistical mechanics*(*A. Carpinteri, F. Mainardi, Eds.*),*Fractal and Fractional Calin Continuum Mechanics*, Springer, Wien, pp. 291–348, 1997.Google Scholar - 27.F. Mainardi, Yu. Luchko and G. Pagnini,
*The fundamental solution of the space-time fractional diffusion equation*, Fractional Calculus and Applied Analysis**4**(2001), 153–1925.MATHMathSciNetGoogle Scholar - 28.M. M. Meerschaert and X. Tadjeran,
*Finite difference approximations for fractional advection-dispersion flow, equations*, J. Appl. Math. and Computing, (2004) in press.Google Scholar - 29.P. J. Olver,
*Applications of Lie Groups to Differential Equations*, Springer, New York, 1986.MATHGoogle Scholar - 30.I. Podlubny,
*Fractional differential equations*, Academic press, San Diego, 1999.MATHGoogle Scholar - 31.A. Saichev and G. Zaslavsky,
*Fractional kinetic equations: solutions and applications*, Chaos**7**(1997), 753–764.MATHCrossRefMathSciNetGoogle Scholar - 32.W. R. Schneider and W. Wyss,
*Fractional diffusion and wave equations*, J. Math. Phys.**30**(1) (1989), 134–144.MATHCrossRefMathSciNetGoogle Scholar - 33.R. Schumer, D. A. Benson, M. M. Meerschaert and S. W. Wheatcraft,
*Eulerian derivation of the fractional advection-dispersion equation*, J. Contaminant Hydrology**48**(2001), 69–88.CrossRefGoogle Scholar - 34.W. Wyss,
*The fractional diffusion equation.*, J. Math. Phys.**27**(11) (1986), 2782–2785.MATHCrossRefMathSciNetGoogle Scholar